People frequently ask for recommendations of great books, thinkers, resources, etc. Below I've collected my personal favorites.
Textbooks
I expect a great deal from a textbook. At a minimum, it should provide the main ideas of a subject in a clear and organized way. Sadly, this is where many textbooks stop, and it disheartens me to see academia continue to use poor books simply because they are "classic" or difficult texts. Given my many cross-country moves for school, I've had to make many hard choices about which books to take and which to donate. I no longer keep books unless they meet the above criterion, and many more, including: having creative and extensive problem sets, interweaving explanations and correcting misconceptions, offering the reader deep intuition behind the results, motivating definitions and theorems, and utilizing visualization as much as possible. I encourage you to get the latest versions of these texts and to toss those books that people love to talk about, but hate to read. If a subject does not appear below, it's because I haven't found a book I love yet.Calculus: Calculus, Early Transcendentals (Stewart)
Real Analysis: Calculus (Spivak)
Abstract Algebra: Abstract Algebra (Dummit & Foote)
Mathematical Statistics: An Introduction to Mathematical Statistics and its Applications (Larsen & Marx), and then Statistical Inference (Casella & Berger)
Machine Learning: An Introduction to Statistical Learning with Applications in R (James, Witten, Hastie, Tibshirani), and then Pattern Classification (Duda, Hart, Stork)
Stochastic Processes/Markov Chains: Introduction to Stochastic Processes with R (Dobrow)